Definition
The normal distribution (or Gaussian distribution) is the symmetric, bell-shaped continuous probability distribution with parameters mu (mean) and sigma (standard deviation). Its density is the famous formula proportional to exp(-(x - mu)² / 2 sigma²).
About 68% of the mass lies within one standard deviation of the mean, 95% within two, and 99.7% within three — the so-called 68-95-99.7 rule. The standard normal has mu = 0 and sigma = 1.
Why it matters
How it works
To work with a normal distribution, standardise: subtract the mean and divide by the standard deviation, converting any normal to the standard normal Z. Tables or calculators then give probabilities like P(Z ≤ z). For example, P(Z ≤ 1.96) ≈ 0.975, which is why 1.96 is the magic number behind 95% confidence intervals.
The normal arises in two main ways. First, mathematically: as the limit distribution of standardised sums, via the central limit theorem. Second, empirically: many physical, biological, and social measurements are approximately normal because they are the cumulative effect of many small independent influences.